Browse › Operator algebras › 299 The Kirchberg–Rørdam character criterion and infinite tensor-power Jiang–Su stability
The Kirchberg–Rørdam character criterion
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Abstract
We prove the Kirchberg–Rørdam character criterion: for every free ultrafilter on ℕ, a nonzero unital separable complex C∗-algebra absorbs the Jiang–Su algebra if and only if its norm central-sequence algebra has no characters. We also prove that the infinite minimal tensor power of every nonzero unital separable complex C∗-algebra without characters is $\mathcal Z$-stable and contains a unital copy of $\mathcal Z$.
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Chris Bruce and Shirly Geffen, C*-selflessness and Z-stability for Thompson's group F (arXiv:2610.28473, 20 Oct 2026). Cites this paper as [42] and proves that the reduced C*-algebra of Thompson's group F is Z-stable, applying the paper's infinite minimal tensor-power corollary to that algebra and embedding the tensor power into it; the authors say they used AI tools, including OpenAI and Anthropic models.
Formalization
- OpenAI, The Kirchberg–Rørdam character criterion and infinite tensor-power Jiang–Su stability: Lean scope note (OpenAI Math Release, lean/docs/299.md, 2026-10-06). Describes the release's Lean formalization of the character criterion (no characters on F_ω(A) iff A ≅ A ⊗_min Z) for every nonzero unital separable complex C*-algebra and every free ultrafilter, and links the comparator statement
CharacterCriterion.lean; the tensor-power corollary is not listed as formalized. [AI-generated per source] [family-level]
Related prior literature mentioned by others
- Eberhard Kirchberg and Mikael Rørdam, When central sequence C-algebras have characters* (arXiv:1409.1395, 2014-09-04; revised 2014-11-28). The AI Proved What? exposition links this paper as the source of the character question and its connection to the Dadarlat–Toms infinite tensor-power question.
Possibly relevant literature (found by the agent, not checked)
- Eberhard Kirchberg and Mikael Rørdam, Central sequence C*-algebras and tensorial absorption of the Jiang-Su algebra (arXiv:1209.5311, 2012; J. Reine Angew. Math.). An earlier Kirchberg–Rørdam study of F_ω(A) and Z-absorption, not cited in the bibliography, that is background for the paper's use of central-sequence criteria in Theorem 1.1.
- Marius Dadarlat, Ilan Hirshberg, Andrew S. Toms and Wilhelm Winter, The Jiang-Su algebra does not always embed (arXiv:0712.2020, 2007). Gives a unital simple nuclear non-type-I algebra with no unital copy of Z, relevant to the paper's point that its tensor-power corollary concerns the infinite tensor power and not the algebra itself.
Digests and expositions
- Open Mathematics Courses / Claude Opus 5.5, Characters of central sequences and Jiang–Su stability (six-lesson online course, undated). This explicitly AI-written exposition follows the manuscript’s finite maximal tensor-power construction through representation-stable labels, a full matrix cone and the dimension-drop algebra, taking the Kirchberg–Rørdam and Dadarlat–Toms structural results as named hypotheses. [AI-generated per source] [family-level]
- AI Proved What?, No characters, then Z-stable? The Kirchberg–Rørdam question (episode 299, October 2026). This AI-assisted video and transcript explain the character criterion, the finite dimension-drop construction and the infinite tensor-power consequence, with examples using matrix algebras and central sequences. [AI-generated per source] [family-level]
News and blogs
- Carlos Toledo, OpenAI’s math release: what it claims beyond what it checks (self-published audit, findings dated 2026-10-10). Its family-299 entry reports that the compared formal statement is narrower than the manuscript’s claims because it does not include the infinite minimal tensor-power Jiang–Su stability conclusion, based on release commit fd4aeeb2. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 299
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:The-Kirchberg-Rordam-character-criterion-September-25-2026,
author = {{OpenAI}},
title = {{The Kirchberg--R\o rdam character criterion}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-Kirchberg-Rordam-character-criterion-September-25-2026/paper.pdf}{OAI:The-Kirchberg-Rordam-character-criterion-September-25-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/299.md. The Lean statements have not been checked by people against the paper's claims.
Family 299: The Kirchberg–Rørdam character criterion and infinite tensor-power Jiang–Su stability
A nonzero unital separable complex C∗-algebra is Jiang–Su stable exactly when its norm central-sequence algebra has no characters, for every free ultrafilter. This answers the Kirchberg–Rørdam character question. Also, the infinite minimal tensor power of every such algebra without characters is Jiang–Su stable, answering the Dadarlat–Toms question.
This is the only paper in this family.